Sorry, I remembered that Nos is plural and Notre is singular. Nos manteaux ou Notre manteau.
Yes but there is a nuance to be made. Notre manteau (our coat) indicates that the subject is plural. I.E. the coat would belong to more than one person... which makes as much sense as spiders eating bread (yes I came across that phrase somewhere in here). Anyway, it is normally used for something that is in common posession: notre voiture (our car), notre maison (our house). My: ''mon/ma'', Nos: ''mes'', Notre: ''our''
As you say, our refers to something in common possession, this means that the subject is plural, if it wasnt you would use 'my' instead of 'our'. The plural version of 'our' in french refers to the object being plural, in this case there are plural coats therefore you use 'nos manteaux' - our coats. The singular version is 'notre manteau' - our coat, which while doesnt make sense in this situation as most people dont share coats is grammatically correct and would make sense with lots of other subjects
It can make sense, Duolingo has probably taught me how to create context around random sentences just as much as it has taught me French. If you were working in groups to make a coat, or if you were speaking on behalf of somebody very close to you (a mother about her son's coat).